Problem 98 - College Algebra Essentials

Question

If \(f(2)=6,\) and \(f\) is one-to-one, find \(x\) satisfying \(8+f^{-1}(x-1)=10\)

Step-by-Step Solution

Answer

The \(x\) that satisfies \(8+f^{-1}(x-1)=10\) is \(7\).

Step 1: Simplification of Expression

Start by isolating the term involved with \(f^{-1}(x-1)\) in the equation \(8+f^{-1}(x-1)=10\). We can simplify this by subtracting 8 from both sides, resulting in \(f^{-1}(x-1)=2\).

Step 2: Using the Property of Inverse Function

Next, we know that for a one-to-one function \(f\), \(f(f^{-1}(y))=y\). Substituting \(y=x-1\) in our equation \(f^{-1}(x-1)=2\) to find \(f(2)=x-1\). We know from the given that \(f(2)=6\), so our equation becomes \(6=x-1\).

Step 3: Finding x

Lastly, solving the equation \(6=x-1\) for \(x\), we can conclude \(x=6 + 1 = 7\).